(Sucker bet). Consider the following gambling game. Player 1 picks a three coin pattern (for example HTH) and player 2 picks another (say THH). A coin is flipped repeatedly and outcomes are recorded until one of the two patterns appears.

Somewhat surprisingly player 2 has a considerable advantage in this game. No matter what player 1 picks, player 2 can win with probability 2=3. Suppose without loss of generality that player 1 picks a pattern that begins with H:

case Player 1 Player 2 Prob. 2 wins 1 HHH THH 7/8 2 HHT THH 3/4 3 HTH HHT 2/3 4 HTT HHT 2/3 Verify the results in the table. You can do this by solving six equations in six unknowns but this is not the easiest way.